The turbulent stress \(\tau \)  is usually dubbed as the Reynolds stress (\(R\) ), which is determined by turbulent pulsations of fluid velocity, e.g. \(\tau _\mathrm {xy} = -\rho \overline{\mathrm {v_x v_y}}\)  for the (xy)-component of the turbulent stress tensor, which is supplementary to the stress tensor of the Reynolds equations system, where \(\rho \)  is the fluid density, \(\mathrm {v_x}\)  and \(\mathrm {v_y}\)  are the x and y-components of the fluid fluctuating velocity, respectively, thus \(\mathrm {V_x} = \overline{\mathrm {V_x}} + \mathrm {v_x}\)  is the whole x-component of the velocity. The Reynolds stresses cannot be derived from the system of Reynolds equations because the latter is unclosed. The usual approach to the turbulent stress is local, after Boussinesq, through the kinematic turbulent (eddy) viscosity coefficient \(\nu _\mathrm {t}\)  and local mean strain: \(\displaystyle \tau _\mathrm {xy} = - \rho \nu _\mathrm {t} \frac{\partial \overline{\mathrm {V_x}}}{\partial y}\) 
in the simple shear, where the gradient \(\displaystyle \frac{\partial \overline{\mathrm {V_x}}}{\partial y}\)  is only non-zero. The turbulent viscosity may be modelled as
\(\displaystyle \nu _\mathrm {t} = \mathrm {v_t} l_\mathrm {m} = l_\mathrm {m}^2 \left| \frac{\partial \overline{\mathrm {V_x}}}{\partial y}\right| \) 
using the concept of Prandtl's mixing length \(l_\mathrm {m}\) , an empirical parameter. In this way Shakura & Sunyaev [1]} prescribed the turbulent fluid viscosity as \(\nu _\mathrm {t} = \mathrm {v_t} l_\mathrm {m} = \alpha c_\mathrm {s} H\)  in the \(\alpha \) -model of accretion disk, where \(\displaystyle \alpha = \frac{\mathrm {v_t} l_\mathrm {m}}{c_\mathrm {s} H}\)  is the viscosity parameter, \(\mathrm {v_t}\)  the turbulent velocity, \(c_\mathrm {s}\)  the speed of sound, \(H\)  the half thickness of the disk. However, the concept of turbulent viscosity does not represent the mechanism of turbulent stress, when large-scale vortices dominate the momentum transport which determines the stress [2]}.
